They are basicly the same apart from HDMI can take the sound. There will be no slow down if you change to HDMI and if you do change to HDMI then there will no quality change from DVI they will both look exactly the same.
DVI and HDMI play very similar roles in that they are both HDCP compatable. HDMI is just a more modern and protected signal, also as klsdivan said, HDMI is also able to carry HD audio signal too. DVI and HDMI will both give you HD and Super HD resolutions, image quality is reliant on the graphics card and what settings you have your games at.